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Mathematics for Physics

53 Gauge

A choice of potentials for the electromagnetic field that is used to fix unphysical freedom while leaving the physical fields unchanged.

Gauge freedom of the potentials. The potentials \(V\) and \(\mathbf{A}\) are not unique. For any scalar function \(\lambda\) one may shift the potentials together without changing \(\mathbf{E}\) and \(\mathbf{B}\). This principle is used to impose an extra condition that simplifies the equations.

A gauge transformation of the potentials is

\[ \mathbf{A}' = \mathbf{A} + \nabla\lambda \]

\[ V' = V - \dfrac{\partial\lambda}{\partial t} \]

where

  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(\lambda\) is an arbitrary scalar function of position and time.
  • \(t\) is time.

The Coulomb gauge. The Coulomb gauge sets the divergence of \(\mathbf{A}\) to zero. This principle is used in magnetostatics and in instantaneous Coulomb problems.

The Coulomb gauge condition is

\[ \nabla\cdot\mathbf{A} = 0 \]

where

  • \(\mathbf{A}\) is the vector potential.

The Lorenz gauge. The Lorenz gauge relates \(V\) and \(\mathbf{A}\) so that both potentials obey wave equations. This principle is used in radiation problems.

The Lorenz gauge condition is

\[ \nabla\cdot\mathbf{A} + \dfrac{1}{c^{2}}\dfrac{\partial V}{\partial t} = 0 \]

where

  • \(\mathbf{A}\) is the vector potential.
  • \(V\) is the scalar potential.
  • \(c\) is the speed of light.
  • \(t\) is time.

Note: Also called a gauge choice. Also called working in a gauge.

53.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — gauge freedom and gauge transformations of \(V\) and \(\mathbf{A}\).
  2. Frankel, T. The Geometry of Physics: An Introduction. Cambridge University Press, 2012. — gauge transformation as a local change of fiber frame.
  3. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Matrix Editions, 2015. — working in a different gauge as a change of bundle coordinates.